The Majorization Theorem and Signless Dirichlet Spectral Radius of Connected Graphs

Abstract

Let $D(G)$ denote the signless Dirichlet spectral radius of the graph $G$ with at least a pendant vertex, and ${\pi }_1$ and ${\pi }_2$ be two nonincreasing unicyclic graphic degree sequences with the same frequency of number $1$. In this paper, the signless Dirichlet spectral radius of connected graphs with a given degree sequence is studied. The results are used to prove a majorization theorem of unicyclic graphs. We prove that if ${\pi }_1⊵{\pi }_2$, then $D(G_1)\le D(G_2)$ with equality if and only if ${\pi }_1={\pi }_2$, where $G_1$ and $G_2$ are the graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with degree sequences ${\pi }_1$ and ${\pi }_2$, respectively. Moreover, the graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with $k$ pendant vertices are characterized.